
No. 95 




THE AVERAGE FORM 



ISOLATE]) SUBMARINE PEAKS, 



THE INTERVAL WHICH SHOULD OBTAIN BETWEEN ])EKP-Si:.\ 

SOUNDINGS TAKEN TO DISCLOSE THE CIlAliACTEK 

OF THE BOTTOM OF THE OCEAN. 



<Lh W. LITTLEHALES, 

V S- HTTDROGRAPHIC OFFICE 



Duder the direction of 
LiMTKXANT RICHARDSOJS CLOVER, U. S. N 
i (ling Uydrographer. 



WASHINGTON : 

GOVERNMENT PRINTING OFFICE. 

isrio. 



No. 95, 




THE AVERAGE FORM 



OF 



ISOLATED SUBMARINE PEAKS, 



AND 



THE INTERVAL WHICH SHOULD OBTAIN BETWEEN DEEP-SEA 

SOUNDINGS TAKEN TO DISCLOSE THE CHARACTER 

OF THE BOTTOM OF THE OCEAN. 



BY 



G. W. LITTLEHALES, 



U. S. HYDROGRAPHIC OFFICE. 



Under the direction of 

Lieutenant RICHARDSON CLOVER, U. S. N., 

Acting Hydrographer. 



WASHINGTON • 

GOVERNMENT PRINTING OFR^CI?. 

1890. >' r 



JUN 7 1906 
D.ofD. 



* : ; .J; 



AN INQUIRY INTO THE AVERAGE FORM OF ISOLATED SUBMARINE 
PEAKS, AND THE INTERVAL WHICH SHOULD OBTAIN BETWEEN 
SOUNDINGS TAKEN TO DISCLOSE THE CHARACTER OF THE BOTTOM 
OF THE OCEAN. 



In a central region in the Atlantic Ocean, though somewhat nearer the 
African than the American coast, in the neighborhood of the remarkable 
volcanic islands known as Madeira and the Canaries, recent bathymetric 
surveys have developed three unsuspected peaks rising abruptly out of 
theocean, which here sinks to^ depth of more than 2,000fathoms. These, 
named Dacia Bank, Seine Bank, and the Salvages, together with Ender- 
bury Island in the central part of the South Pacific Ocean, and the un- 
named shoal within late years developed by the United States ships 
Tuscarora and Banger^ lying to the westward of San Francisco in the 
track of commerce, are taken as the basis of discussion, as they have been 
more fully explored than other formations of this class. The deep-sea 
soundings which have been observed in each of these localities are shown 
on the accompanying charts. Among eminent authorities there is great 
diversity of opinion concerning the proper interval between deep-sea 
soundings taken to develop the character of the bottom of the ocean and 
in searching for reported shoals. With regard to the relative slopes 
of land and submarine peaks the same diversity of opinion exists; 
some well-informed men asserting that land peaks, although perhaps 
originally of the same general slopes as those beneath the surface of the 
ocean, have been diminished in steepness by the gradual weathering 
and transference of the higher portions toward the base, while others 
maintain that as peaks formed in the ocean have the weight of the 
water to bear in addition to their own weight, their slopes must be less 
steep than those formed on land. 

This inquiry is undertaken with a view of fixing the ideas of naviga- 
tors as to the proper interval between deep-sea soundings taken to 
develop the existence of important changes in the bed of the ocean, and 
also with a view of affording some means of making comparisons be- 
tween the slopes of land and submarine peaks by deducing the equation 
to the curve which, by revolution around a vertical axis, would gen- 
erate the average of the surfaces of the submarine peaks above men- 
tioned. 



Theoretically the shape of ao isolated submarine formation would be 
that of a solid of revolution in which the crushing strength of any sec- 
tion is equal to the combined weight of the portion of the formation 
above that section and of the superincumbent body of water. 

Let y denote the radius of any section, and x its distance from the 
top of the formation. 

Let K denote the co-efficient of crushing strength of the material of 
the formation, d the weight of a unit of its volume, and d' the weight of 
a unit of volume of sea- vvater. 

Assuming that the top of the formation just reaches to the surface 
of the ocean, 

TtbCy^dx = the weight of the formation above any section whose 
distance from the top is j?, 
2itd'fy.x.dy = the pressure of the water upon the formation above any 
section whose distance from the top is x, 
TtKy'^ — the strength of any section to resist crushing. 
Then 

Tz^fyHx-\-2itb'fy.x.dyz=7iKfJ^id . . . . (1) 

in which is a constant representing the excess of crushing strength 
in any section above what is necessary to withstand the pressure caused 
by the weight of the formation and the weight of the superincumbent 
body of water. 
By differentiation, equation (1) becomes 

ndy'^dx -f 27r(5'2/ ,x,dy = 27: Ky . dy 
or 

d dx dy 



2 (K - d'x) y 
d dx _ dv 

By integration, equation (2) becomes 

K 



(2) 



y log (x-^^ = log y 

l0g(^^-^j= ylOgI/ 



K flog. 



^^K^^^'iog. (3) 



As there are no well-determined data for the density and crushing 
strength of the materials which compose these formations, it seems 
most practical to form conditional equations by inserting observed val- 
ues of a? and y in equation (3) written in the form 

x = A + B£'°^% 

and to find the values of the constants A and B by the Method of Least 
Squares. The following observed values, in which y is expressed in 
nautical miles and x in fathoms, are taken from the accompanying 
charts : 

DACIA BANK. 

^ = + 310 10' ;. = + 130 40^ 

y= 1, 2, ^, 3, 4, 5, 74, 23i, 26. 
X = 335, 619, 757, 844, 1102, 1189, 1386, 1592, 1961. 

SEINE BANK. 

^ = + 330 50^ / = + 140 20' 

2/ = 1, 2, 3, 6, 0, 13, 14, 14. 
X = 289, 845, 1149, 1769, 2210, 2250, 2190, 2325. 

THE SALVAGES. 

<p = -\- 30O 05' ;. = + 150 55' 

y= 1, 7^, 12. 

X = 778, 1688, 1827. 

ENDERBURY ISLAND. 
^ = -30IO' ^=4- 171O10' 

y= li, 3i, 114, 17. 
X = 880, 1991, 2370, 2835. 

SHOAL IN NORTH PACIFIC OCEAN. 

c> = + 320 55' ;. = +1320 30' 

2/= 1, 3, 6, 12. 
X = 388, 1045, 1870, 2282. 



By inserting- tbese values we have the following conditional equa- 
tions : 



(.1) 335.000 = 


V. vuv 

= A 4- 2. 718281828 

0.301 


B == A + 1. 0000000 


(2) 619. 


0. 398 


1.3512 


(3) 757. 


0.477 


1.4888 


(i) 844. 


0. 602 


1.6112 


(5) 1102. 


0.699 


1. 8258 


(6) 1189. 


0.875 


2. 0117 


(7) 1386. 


1.371 


2. 3988 


(8) 1592. 


1.415 


3. 9393 


(9) 1961. 


0.000 


4. 1165 


(10) 778. 


0. 875 


1 0000 


(11) 1688. 


1. 079 


2. 3988 


(12) 1827. 


0. 000 


2. 9418 


(13) 289. 


0.301 


1. 0000 


(14) 845. 


0.477 


1. 3512 


(15) 1149. 


0.778 


1. 6112 


(16) 1769. 


1.146 


2. 1771 


(17) 2190. 


0.954 


3. 1456 


(18) 2210. 


1.146 


2. 5961 


(19) 2325. 


1.114 


3. 1456 


(20) 2250. 


0. 000 


3. 0465 


(21) 388. 


.0.477 


1. 0000 


(22) 1045. 


1.079 


1. 6112 


(23) 2282. 


0.778 


2. 9418 


(24) 1870. 


0. 176 


2. 1771 


(25) 880. 


0. 544 


1. 1924 


(26) 1991. 


1.061 


1. 7229 


(27) 2370. 


1.230 


2. 2893 


(28) 2835. 




3.4213 


40766. 000 


28 A + 60, 5132 B 



From the above conditional equations the foUowing normal equa- 
tions are formed : 

40766.000 = 28.000 A + 60.513 B 
102321.027 = 60.513 + 152.932 
From which 

B= + 641.8396 

A = + 68.79.-5 

Therefore the equation to the average of the outlines of the vertical 
sections of the submarine peaks under discussion is 

jc = 4- 68.7985 + 641.8396 e '"' '■' , 



from which: 

2/ =0.33, 1, 2, 3, 4, 5, 6, 9, 10, 12, U. 
X = 465, 711, 936, 1103, 1241, 1360, 1466, 1733, 1813, 1937, 2088. 

Both theory and actual examples show that there is a close agree- 
ment in general form and slopes between land and submarine peaks, 
and, while it is noticeable that those which are submerged are some- 
what less steep than those on land, it is apparent that the latter have 
been formed with sufficient strength to resist the extra pressure to 
which they would be subjected if submerged.* 

It is also shown that isolated formations occupying comparatively 
limited areas at the bottom can and do occur in deep water, and we are 
able to assign at once a maximum interval which should obtain be- 
tween soundings taken to develop the general character of the bottom 
of the ocean or in searching for reported ocean shoals. Soundings 
taken 40 miles apart in a depth of 2,500 fathoms do not prove anything 
with certainty, except the depth at the points where they are taken, 
because it is possible with this interval to pass entirely over a forma- 
tion rising from the bottom to within a short distance of the surface. 
The minimum radius at the bottom which a dangerous ocean shoal can 
have must vary directly with the depth, but on the average in the deep 
sea it may be stated as 10 miles. An interval of 10 miles coupled with 
an interval of 2 miles would be sufficient for general development, and 
would prove with certainty the existence or absence of any formation 
rising close to the surface. Of all the possible ways in which a 10-mile 
interval could lie with reference to a submerged peak, that which would 
be most advantageous for a prompt discovery is the condition in which 
one end of the interval is at the bottom of the slope and the other near 
the apex, and that which would be least advantageous is the condition 
in which the interval is bisected by the position of the apex. In the 
latter case there would be nearly equal soundings at both ends, but the 
soundings at the ends of the adjacent 2-mile intervals would immedi- 
ately disclose the slopes. Whenever lines of deep-sea soundings are 
run, it is better and gives more useful information to place the sound- 
ings at alternate long and short distances apart. Thus it is better to 
take soundings along such a line at alternate intervals ot 10 miles and 
2 miles than at regular intervals of 6 miles. In most cases the 10-mile 
soundings will give as much information as the 6-mile soundings, but 
the 2-mile soundings at an average distance of 12 miles apart give defi- 
nite information about the actual nature of the bottom, and not merely 
its average character. 

* See Mr. G. F. Becker's paper on the form of volcanic cones, in the Am. Jour, of 
Science, 1885, p. 283. 



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